Energy from Wind

Power• Power: Rate at which energy is delivered

Power = Energy Time

• Measured in Watts (W), kilowatts (kW), or horsepower

• Power is an instantaneous quantity• Power does not accumulate• Think gallons per minute

Energy• Energy: Ability to do something

• Measured in kilowatt Hours (kWhrs)• Why?

– Since Power = Energy/Time,

then Power Time = Energy

• Energy does accumulates over time• Think gallons• Gallons = (gallons/min) minutes

PowerkW

(kilowatts)

EnergykWh

(kilowatt hours)

Think gpm

Think gallons

Wind Resource• At any instant, the only question that makes

sense is “What’s the power of the wind?”• Answer depends on 2 quantities

– Instantaneous wind speed, v– Air density, , which depends on

• Elevation• Temperature• Weather• At sea level and 77F (standard conditions), air density =

1.225 kg/m3

• At 5,000 ft elevation, is ~16% less than at sea level

Power Density of the Wind• Power Density: P/A

P/A = ½ v3 (in W/m2)

• Example: Suppose the wind speed is 8.0 m/s, and air density is 1.0 kg/m3, then

P/A = ½ (1.0 kg/m3)(8.0 m/s)3 = 256 W/m2

– For each square meter of area, there are 256 W of power– Use Metric Units!– If wind speed doubles, power density increases by 8

Swept Area• The single most important parameter of a wind

turbine is its rotor’s swept area

A

Power of a Wind Turbine• The power of a wind turbine is

P = ½ v3 A CP

A: swept area of rotorCP: rotor efficiency

• Example: A 2.5 m diameter turbine with a 25% efficient rotor in our 8.0 m/s wind will have

P = ½ (1.0 kg/m3)(8.0 m/s)3 [ (2.5 m/2)2](0.25)

= 314 W

How NOT to estimate energy in the wind

• How much energy can this turbine produce? • Need a constant wind speed and time• Example: If the wind speed is a constant 8.0

m/s, then in 1 month our turbine will produce– (314 W)(30 days)(24 hrs/day) = 226 kWhrs/month– The average home in NC uses around 850

kWhrs/month• The wind speed is not constant

10 Minute Wind Data

0 10 20 30 40 500

2

4

6

8

Fre

qu

ency

(%

)

Probability Distribution Function

50WS HI (mph)

Actual data Best-f it Weibull distribution (k=2.04, c=15.96 mph)

Wind Speed Distributions

Using the Annual Average Wind Speed to Calculate Energy Production is Problematic

• Using the average Annual wind speed will under estimate energy production because of the cubic relationship between wind speed and power.

• Need to cube each 10 minute wind speed• The average of the cubes is greater than the

cube of the average

Cube of Average vs Average of Cubes for site with 6.5 m/s average annual wind speed

• Cube of the Average– Class 3 site @ 30 meters

= 6.5 m/s– P/A = .6125 x 6.53

– P/A = 168 watts/m2

Too Low

• Average of CubesP/A of 10.0 m/s = 612P/A of 5.0 m/s = 76.56P/A of 4.6 m/s = 59.6

19.6/3748.16/3

6.5 m/s 249 w/m2

Energy Pattern Factor (EPF) = Average of cubes / cube of average = 249 / 168 = 1.48

10 minute datamph20 std dir F

10/1/2006 0:00 1.00 0.6 0 5010/1/2006 0:10 1.00 0.5 202 5010/1/2006 0:20 3.10 1 270 5010/1/2006 0:30 3.60 0.9 248 5010/1/2006 0:40 4.00 1.6 225 5110/1/2006 0:50 6.70 2.4 225 5310/1/2006 1:00 5.50 2.1 202 5410/1/2006 1:10 8.90 2.5 202 5410/1/2006 1:20 8.50 2.2 202 5510/1/2006 1:30 7.50 2.8 225 5510/1/2006 1:40 5.40 1.9 225 5510/1/2006 1:50 4.50 1.9 225 5510/1/2006 2:00 4.00 2 270 55

Average of Cubes is Greater than Cube of Average

Time Stamp Speed (mph) m/s P/A1/1/2006 15:50 7.6 3.39 23.921/1/2006 16:00 8.2 3.66 30.051/1/2006 16:10 9.2 4.11 42.441/1/2006 16:20 10.5 4.69 63.091/1/2006 16:30 10.6 4.73 64.911/1/2006 16:40 9.8 4.38 51.291/1/2006 16:50 10.3 4.60 59.551/1/2006 17:00 10.6 4.73 64.911/1/2006 17:10 12.4 5.54 103.901/1/2006 17:20 10.9 4.87 70.571/1/2006 17:30 11.4 5.09 80.741/1/2006 17:40 12.2 5.45 98.96

average speed 4.60

P/A of Average 59.69 watts/m2Average of Cubes 62.86 watts/m2

Energy Pattern Factor

• Average of Cubes divided by Cube of Average• 62.86 / 59.69 = 1.05• EPF = 1.05• Typical EPF = 1.9• Multiply power density calculated from

average annual wind speed by 1.9 to get more accurate average annual power density

Estimating Average Annual Power Density from Annual Average Wind Speed

• What would be a reasonable estimate of an annual average power density when the average annual wind speed was 12 mph (5.35 m/s) and elevation was 4,000’

• Annual Average P/A = ½ Density x V3 (in meters/sec) x 1.9• AA P/A of 12 mph = ½ (1.225 x .88) x 5.353 x 1.9• AA P/A of a 12 mph wind at 4,000’ = 156 watts/m2

Air Density Changes with Elevation

Density Change with Elevation

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

10,000

70 75 80 85 90 95 100

Density Change Compared to Sea Level, %

Ele

vati

on

, ft

Swept Area Method of Estimating Energy Production (AEO)

• AEO = (Average annual power density x 1.9) x area of rotor (m2) x efficiency x hours/year

Swept Area• Power is directly related to the area intercepting the

wind• Doubling the swept area will double power available

to it• Nothing tells you more about a wind turbines

potential than area swept by rotor• Area = πr2 or πd2/4• Relatively small increases in blade length produce

large increase in swept area• Doubling diameter will quadruple swept area

Credit: Paul Gipe

Swept Area

A = Pi D2 / 4

1 m = 3.3 ft

Area = πr2

Swept Area of Bergey XL.1

• Bergey XL.1 has three blades each 4’ long and a rotor diameter of 8.2’

• 8.2’ / 3.28 (ft/m) = 2.5 meter diameter

• Radius = 1.25 meter• Area = πr2

• Area = πr2 = π 1.252 = 4.9 m2

Power Intercepted by Bergey XL1 with 4.9 m2 of Wind Power at 4,000’, 00, in 7 m/s wind

• Power = ½ density x area x velocity3

• Power = ½ (1.218 kg/m2) x 4.9 m2 x 73

• Power = .609 x 4.9 m2 x 73

• Power = .609 x 4.9 x 343• Power = 1,023 watts

Estimating Annual Energy Output of XL.1 with Swept Area Method @ class 3 site; 6.5 m/s @ 5,000’

• AEO in watts = Annual Average P/A x Swept Area x efficiency x hours per year

• AEO = (1/2 air density) x (v3) x (1.9) x 4.9 x .20 x 8760

• AEO = ½ (1.225 x .860) x (6.53) x 1.9 x 4.9 x .20 x 8760

• AEO = 2,359 Kwh

Power Curve Method or Method of Bins

2 Things Needed

Need to know (or approximate) your wind distribution

Power Curve of turbine

Wind Distribution• Wind is known to follow a Weibull distribution

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240

500

1000

1500

2000

2500

3000

Distribution of Wind Speeds

Frequency

Wind Speed (m/s)

# o

f O

ccu

rren

ces

Wind Distribution• Wind is known to follow a Weibull distribution • =WEIBULL(c, k, vavg)• Rayleigh Distribution if k=2

Credit: Paul Gipe

Wind Speed Distributions

k = 2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Pro

b.

den

sity

k = 3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Pro

b.

den

sity

k = 1.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Pro

b.

den

sity

• Wind is empirically known to follow a Weibull probability distribution

• Weibull curve: has shape parameters: c & k• Average k in US: k = 2 (Raleigh distribution)

Method of BinsWind Distribution: From your logger!

Power Curve• The turbine’s manufacturer will provide you

with its power curve

Bergey XL.1

Whisper Power Curves

Utility Scale Power Curve (GE)

Method of BinsPower Curve (kW)

Wind Distribution (hrs)

AEO (kWhrs)H

ours

Ener

gy (k

Whr

)

Method of Bins• Calculate Energy = Power Time for each wind

speed bin• Sum ‘um up!

Charts from Manufacturer